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A307212 a(n) is the Narumi-Katayama index of the Lucas cube Lambda(n).

%I A307212 #19 Apr 02 2019 14:34:42
%S A307212 0,2,3,256,38880,1289945088,42855402240000000,
%T A307212 605828739547255327948800000000,
%U A307212 13263549731442762279026688000000000000000000000000000,1334793240853871268746431553848403294648071618560000000000000000000000000000000000000000000
%N A307212 a(n) is the Narumi-Katayama index of the Lucas cube Lambda(n).
%C A307212 The Lucas cube Lambda(n) can be defined as the graph whose vertices are the binary strings of length n without either two consecutive 1's or a 1 in the first and in the last position, and in which two vertices are adjacent when their Hamming distance is exactly 1.
%C A307212 The Narumi-Katayama index of a connected graph is the product of the degrees of the vertices of the graph.
%H A307212 I. Gutman and M. Ghorbani, <a href="https://doi.org/10.1016/j.aml.2011.12.018">Some properties of the Narumi-Katayama index</a>, Applied Mathematics Letters, Vol. 25, No. 10 (2012), 1435-1438.
%H A307212 S. Klavžar, M. Mollard and M. Petkovšek, <a href="https://doi.org/10.1016/j.disc.2011.03.019">The degree sequence of Fibonacci and Lucas cubes</a>, Discrete Mathematics, Vol. 311, No. 14 (2011), 1310-1322.
%e A307212 a(2) = 2 because the Lucas cube Lambda(2) is the path-tree P_3 having 2 vertices of degree 1 and 1 vertex of degree 2; consequently, the Narumi-Katayama index is 1*1*2 = 2.
%p A307212 G := (1+(1-y)*x+x^2*y^2+(1-y)*x^3*y-(1-y)^2*x^4*y)/((1-x*y)*(1-x^2*y)-x^3*y):
%p A307212 g := expand(series(G, x=0, 40)): T := (n, k) -> coeff(coeff(g, x, n), y, k):
%p A307212 a := n -> mul(k^T(n, k), k=0..n): lprint(seq(a(n), n=1..10));
%Y A307212 Cf. A307157, A307181, A307307.
%K A307212 nonn
%O A307212 1,2
%A A307212 _Emeric Deutsch_, Mar 28 2019